Senator John Thune of South Dakota, the No. 2 Republican, told his hometown newspaper, The Argus Leader, that he was “confident Congress will be able to find common ground on the so-called red flag issue.” Senator Lindsey Graham, Republican of South Carolina, has already proposed legislation that would offer federal grants to states to help them enact and enforce red flag laws, also known as “extreme risk protection orders,” which are intended to restrict potentially dangerous people rather than dangerous weapons.

And Mr. McConnell has asked three committee chairmen to “reflect on the subjects the president raised” and hold bipartisan talks of “potential solutions.”

Red flag legislation also appears poised to move in the House. The Judiciary Committee was consulting with its members on Tuesday about whether to briefly return to Washington from a six-week recess to advance a red flag bill and other gun-related legislation, according to an aide to the committee.

But it is not clear how Democrats will proceed. Some House liberals want still more measures, such as a ban on military-style assault weapons and high-capacity magazines, which directly fuel mass shootings. A House Democratic leadership aide suggested that a red flag bill passed out of the Senate would ideally be attached to tougher House bills to force negotiations between the two chambers.